I want to know how to explain the Intermediate Value Theorem in its
most simple form: If f is a continuous function on the closed interval
[a,b] and N is any number between f(a) and f(b), there must be a
number c in (a,b) such that f(c)= N.

Any route traveling from (0,0) to (1,1) going only north and east will
cover a total distance of 2 units. But the straight line distance from
(0,0) to (1,1) is sqrt(2) units. It seems that if I think of a
staircase connecting the two points and let the stairs become
infinitely small, the limit of the north/east route distance should
converge to sqrt(2). But it doesn't! What's going on here?

(1) The cost of goods and services in an urban area increased 1.5% last
month. If the rate continues, what will be the annual rate of increase?
(2) Suppose a certain population increases by 30% every 10 years. By what
percent does it increase each year?

Can you help me show, with and without calculus, that the geometric figure of a maximum area and given perimeter is a circle? What are the dimensions of a triangle with perimeter p that encloses the maximum area?

For a limit to exist, the right side and left side have to equal each
other. Say you have a limit such as "lim as x->4 for function (3x-4). To
see if this function exists, you have to check it on both sides. Would
you plug in 3 for the left and for the right?